I think the connection works something like this:
One of the best ways to prevent STIs is regular screening, whether or not you're symptomatic. Reduced services means people test less often, or only when they have symptoms that they notice. This means when someone acquires an STI they are more likely to transmit it to others for a longer period of time.
Beyond that, cutbacks in services means less education about prevention, fewer free condoms, fewer opportunities for people to have contact with medical professionals with expertise in sexual health, etc.
Here's another article that focuses more squarely on the connection between STIs and cuts to care and prevention programmes: http://www.npr.org/sections/health-shots/2016/10/20/498719092/std etc ...